Firstly, regarding voting habits;
I don't know about others but, similarly to @GarethMcCaughan (though I think I may be less miserly), I think of not voting as a middle of the road option.
I base my decision (up, down, neither) on:
1) whether a question is well written and easy to understand,
and 2) whether the answer is fitting or satisfying
So, because the answer is a very important part of the complete puzzle, I vote questions up almost exclusively after an answer has been accepted so that I can decide whether or not the question made sense, baring in mind that the correct answer to a puzzle (unless you're Edgar Allan Poe) is determined by the asker before they even ask!
Answers however are different! We typically read a lot more answers than we do questions, and I do not restrict my up-votes to accepted or correct or even partially-correct answers, but rather to answers that are helpful (or sometimes just very clever or funny to be honest). So sometimes I may up-vote an answer and then leave - never to return - without up-voting a question simply because I didn't find out whether this was the intended answer of the OP, but this doesn't necessarily mean that I think this was a 'valueless puzzle with a valuable solution'.
Secondly, regarding my answers;
I answer questions that I have not voted on abiding by the same logic; I usually can't determine whether a puzzle is good or not without knowing the answer. I have not been able to determine the value of the puzzle, but I have answered it. So in this respect, as you put it, I may 'not appreciate many puzzles they find worthwhile to answer'.
Thirdly, regarding whether 'it is good for Puzzling when a solution is posted by someone who is not interested enough to vote for the puzzle';
my answer would be yes, because you'd be surprised the amount of times I've wondered whether I should guess at an answer, where I've been stumped "but actually maybe its... NO! Surely not! That would be terrible! I so hope that's not the answer... I guess there's only one way to find out."
So... even assuming that any puzzle that I do not dislike would get a +1, even if I've bothered to provide an answer and even if that answer has been accepted, I may then decide (and it has happened) "Hey, that wasn't satisfying at all. I guess it wasn't just poor wording after all. I'm actually disappointed that this was the answer they were looking for. I really just wanted to rule it out and maybe help move things along."
Also, I've up-voted answers to questions that were probably posted for similar reasons! It helps to stop you sitting staring at a question hoping to fathom some hidden grain of information that will make sense of, and entirely justify, a question that, in a state of not knowing, could be really excellent or actually a bit poo.
And lastly, an answer to a question you didn't actually ask;
Would it be good for PSE if there was a culture wherein people valued questions more; enough that they might keep track of questions that they've looked at and wondered about (and maybe up-voted an answer or two), and eventually return to that question to see what the accepted answer turned out to be so that they can up-vote it if they deem it appropriate to do so? maybe and maybe I'll try to be more like that.
Thinking about it now, I think I remember reading a badge description that specified that a certain minimum percentage of total up-votes must be on questions, so it can't be the first time someone's wondered about this sort of thing!